Piecewise hereditary algebras

We call a finite-dimensional k-algebra A piecevise hereditary if Db(A) is triangle-equivalent to Db(kΔ) for some finite quiver Δ without oriented cycle. We recall that Δ is uniquely determined up to the relation ∼ introduced in I. 5.7. In this section we present some general facts about piecewise hereditary algebras. But first we need to recall some elementary facts for hereditary finite-dimensional k-algebras.

LEMMA. Let B be a hereditary finite-dimensional k-algebra and let X1,X2,X3 be B-modules. Suppose that f: X1 → X2 is subjective and that g: X2→ X3 is injective. Then there exists a module Y and linear maps h1: X1 → Y and h2: Y → X3 such that

is exact.

Proof. Consider the following exact sequence

Since B is hereditary, Ext1B(X3/X2,f) is surjective. Let

be a preimage of (*) in Ext1B(X3/X2,X1). Then we obtain the following commutative diagram of exact sequences

with h1 injective and h2 subjective. By construction we have that

is exact.

LEMMA. Let B be a hereditary finite-dimensional k-algebra and let X,Y be indecomposable B-modules. Suppose that Ext1B(Y,X)= 0. Then 0 ≠h ∈ HomB(X,Y) is either injective or surjective.

Proof. Let 0 ≠h ∈ HomB(X,Y). Let X → Z → Y be a factorization of h with f surjective and g injective. By 1.2 we obtain an exact sequence

By assumption this sequence splits. By Krull-Schmidt we infer that Z is isomorphic to X or isomorphic to Y.

We also note the following immediate consequence for an indecomposable module X over a hereditary finite-dimensional k-algebraB.